projective geometry: Sylvester-Gallai Theorem

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A mathematician, named Phlexicon, contacted me today, about a error in one of my proof in my learning note of Introduction to Real Projective Plane .

We chatted on Skype for about 40 min. He showed me, how my version of the proof on Sylvester-Gallai Theorem was wrong. Though, it's been 13 years since i wrote the proof, so, i couldn't seriously understand it without spending few days reviewing the stuff. Though, he convinced me he's right. I remember, when i was studying it, the proof given in the text seems a bit complex and convoluted, and i thought to myself that it should be done with a simple induction proof. After i did a proof, i wondered a bit why the problem went without a proof for some 40 years. This is back in 1996, and the internet only started, without all the blog and feedback and social networks, and i didn't put my notes online until 2004, and didn't show it to anyone.

He pointed out the Wikipedia article: Sylvester-Gallai theorem. That is a wealth of info.

Phlexicon also showed me a interesting elementary geometry problem. You might try to show it to your kids in highschool. Here's the problem:

Suppose there's a pyramid (as in Egyptian pyramid, with a square bottom), such that each of the triangle faces are Equilateral triangle. Let's call this pyramid p4. Now, let's say there's a tetrahedron, which is also a pyramid but with the base being a equilateral triangle. Let's call this p3. The question is, what is the ratio of volume of p3 and p4. You are to solve this problem by insight, and you are not allowed to use algebraic formulas.

Phlexicon said that this problem can be solved by insight, with 2 “aha” realizations. I thought about it for 20 min yesterday but haven't seen it yet.

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